ALEXANDRIAN SCIENCE. 



37 



matics. The theorem of the Lunula (or Lunes) of Hippocrates is an 

 elegant demonstration of the strict equality of the area of two crescent- 

 shaped figures to that of a triangle, and it is so simple that even the 

 unmathematical reader who has noticed the theorem of Pythagoras 



FIG 10. 



FIG. 9. 



(p. 13) and that of Thales (p. 8), mentioned in the preceding 

 chapter, will have no difficulty in following the demonstration. He 

 must first preceive that if circles were inscribed in each of the squares 

 in Fig. 4, the circle in the square opposite to the right angle would 

 have an area equal to those of the other two circles put together also 

 that the like would be true of the halves of the semicircles. In Fig. 

 10, A F B G c is the semicircle on A c; A D B is that on A B; EEC 

 is that on B c ; and the area of the first semicircle is equal to that of 

 the second and third together. If from these equals we take away 

 the common parts (denoted by the shaded spaces), the remainders 

 will be equal ; thus the areas of the lunulce A F B D, B G c E, will to- 

 gether equal that of the triangle ABC. 



Euclid extended the resources of geometrical reasoning by his adop- 

 tion of the method of exhaustions as it was termed, which is only an 

 application of the general principle of limits. The geometrical meaning 

 of this term may be easily understood. If a regular hexagon (Fig. 9) 

 be inscribed in a circle, it is obvious that the area of the hexagon is less 

 than that of the circle, and also that the circumference of the circle 

 has a greater length than the sum of the sides of the polygon. Now, 

 suppose that by drawing straight lines between the extremities of each 

 side of the hexagon, and a point in the circumference half-way between 

 them, we inscribe a twelve-sided figure in the circle, it will be equally 

 obvious that the area and perimeter of this figure will be greater than 

 those of the hexagon, but yet less than those of the circle. By doubling 

 the number of sides in the polygon, we may approximate yet more 

 closely to the circle, and by sufficiently increasing that number we may 

 arrive at a rectilinear figure as nearly coinciding with the circle as we 

 please. The circle is therefore said to be the limit of the inscribed 

 polygon; that is, the boundary to which the latter continually ap- 



